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To solve this problem, we’ll use the information provided and apply the principle of set theory. Let’s define the following:

- \( n = 70 \): total number of students
- \( n(CA) = 51 \): number of students who opted for College Algebra (CA)
- \( n(BM) = 55 \): number of students who opted for Business Mathematics (BM)
- \( n(CA \cap BM) = 48 \): number of students who opted for both CA and BM

We need to find:
1. The number of students who opted for neither course.
2. The number of students who opted for CA but not BM.

### Step 1: Find the Number of Students Who Opted for Either CA or BM
Using the formula for the union of two sets:
\[
n(CA \cup BM) = n(CA) + n(BM) - n(CA \cap BM)
\]
Substitute the values:
\[
n(CA \cup BM) = 51 + 55 - 48 = 58
\]

So, 58 students opted for either CA or BM (or both).

### Step 2: Find the Number of Students Who Opted for Neither Course
\[
\text{Number of students who opted for neither course} = n - n(CA \cup BM)
\]
\[
= 70 - 58 = 12
\]

Therefore, 12 students did not opt for either course.

### Step 3: Find the Number of Students Who Opted for CA but Not BM
Using the formula for students who opted only for CA:
\[
n(CA \text{ only}) = n(CA) - n(CA \cap BM)
\]
Substitute the values:
\[
n(CA \text{ only}) = 51 - 48 = 3
\]

So, 3 students opted for CA but not BM.

### Summary
- **Number of students who opted for neither course:** 12
- **Number of students who opted for CA but not BM:** 3

### Venn Diagram
To complete the problem, we will draw a Venn Diagram. I’ll provide a description for creating one:

1. Draw two intersecting circles, one for \( CA \) and one for \( BM \).
2. In the intersection (center) of the circles, write \( 48 \) to represent students who opted for both courses.
3. In the section of \( CA \) only (left side outside the intersection), write \( 3 \) for students who opted only for \( CA \).
4. In the section of \( BM \) only (right side outside the intersection), write \( 7 \) because \( 55 - 48 = 7 \).
5. Outside both circles, write \( 12 \) to indicate students who opted for neither course.

Would you like any further details on any part of this solution?

### Related Questions
1. How many students opted only for Business Mathematics?
2. What percentage of students opted for College Algebra?
3. If 10 more students opted for neither course, how would that affect the solution?
4. Can this problem be solved with a different approach, such as using a different set notation?
5. How can set theory be applied to more complex scenarios involving three or more sets?

**Tip:** For similar problems, always start with set notation to clearly organize the information you have.

A class of 70 students opted two courses as follows: 51 opted the course College Algebra (CA), 55 opted the course Business Mathematics (BM), and 48 opted both the courses CA and BM. How many opted neither of the courses? How many opted course CA but not the course BM?

Solve a set theory problem involving two courses: College Algebra (CA) and Business Mathematics (BM). Learn how to use Venn diagrams to find students who opted for neither or only one course. This problem demonstrates the principle of inclusion-exclusion, suitable for students in Grades 9-11.

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